Search

In the context of large-scale linear programs solved by a column generation algorithm, we present a primal algorithm for handling the master problem. Successive approximations of the latter are created to converge to optimality. The main properties are that, for every approximation except the last one, the cost of the solution decreases whereas the sum of the variable values increases. Moreover, the minimum reduced cost of the variables also increases and converges to zero with a super-geometric growth rate.

Given a linear program? ( L P ) with m constraints and n lower and upper bounded variables, any solution x 0 to L P can be represented as a nonnegative combination of at most m + n so-called weighted paths and weighted cycles, among which at most n weighted cycles. This fundamental decomposition theorem leads us to derive, on the residual problem L P ( x 0 ) , two alternative optimality conditions for linear programming, and eventually, a class of primal algorithms that rely on an Augmenting Weighted Cycle Theorem.